For sharing the Eon Ticket:
The ticket is shared by mixing records at the Record Corner. The person who has the ticket should be Player 1 (smaller plug of the cable).
If it doesn't work, here are things to troubleshoot:
- The game with the Eon Ticket has to have gotten it as an original event and not from another Player.
- Save files may be limited on the number of times they can send a ticket via record mixing - seems to have been the case for me after sharing twice, but this could be user error on the last attempt.
- Make sure your Game Link Cable (Wireless Adapters won't work for Ruby/Sapphire) is for GBA and not Game Boy Color. Both fit the socket, but GBA will have tabs to fill the lower notch. You'll probably already know if it's the right kind if you've used it to trade between games before, though.